Asymptotic agreement of moments and higher order contraction in the Burgers equation

The purpose of this paper is to investigate the relation between the moments and the asymptotic behavior of solutions to the Burgers equation. The Burgers equation is a special nonlinear problem that turns into a linear one after the Cole–Hopf transformation. Our asymptotic analysis depends on this...

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Bibliographic Details
Published in:Journal of Differential Equations 2010-05, Vol.248 (10), p.2417-2434
Main Authors: Chung, Jaywan, Kim, Eugenia, Kim, Yong-Jung
Format: Article
Language:English
Subjects:
Burgers equation
Cole–Hopf transformation backward moment
Complex Gaussian
Heat equation
Long-time asymptotics
Truncated moment problem
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Summary:The purpose of this paper is to investigate the relation between the moments and the asymptotic behavior of solutions to the Burgers equation. The Burgers equation is a special nonlinear problem that turns into a linear one after the Cole–Hopf transformation. Our asymptotic analysis depends on this transformation. In this paper an asymptotic approximate solution is constructed, which is given by the inverse Cole–Hopf transformation of a summation of n heat kernels. The k-th order moments of the exact and the approximate solution are contracting with order O ( ( t ) k − 2 n − 1 + 1 / p ) in L p -norm as t → ∞ . This asymptotics indicates that the convergence order is increased by a similarity scale whenever the order of controlled moments is increased by one. The theoretical asymptotic convergence orders are tested numerically.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2010.01.006